The question confuses the actual turn of events and the characters' intentions. Bourne didn't insist, he offered her the choice because he recognized she doesn't want to leave him.
According to the script, Marie sees this relationship as more than just a gig, Jason doesn't.
He turns back. She's staring at him.
Sorry. The money, right?
Before she can say anything, he's digging in the backpack.
He pulls out another stack of hundreds. Hands it over. She
takes it. It's not what she wanted, but she's used to being
disappointed. Fighting it.
He thanks for the ride and wants to go. Until he finally gets it.
Thanks for the ride.
Silence. That moment. He focuses. Getting it.
Getting that she doesn't want this to end. He realizes the danger and wants her to realize it too. She still wants to go with him because she thought he would forget her the moment they parted ways.
Look, I don't know what's up there.
You got me pretty fucking curious.
Look, you could come up. Or you
could wait if you want. I could go
check it out. You could wait.
(hide the pain)
With you, I mean, you'd probably
just forget about me, right?
At this moment he warned her twice she shouldn't go with him. Yet she ignored the warning and made her decision. He agrees.
How could I forget about you?
You're the only person I know.
MARIE smiles. We've never seen it before. Worth waiting for.
At this moment the relationship becomes romantic.